A253916 -id:A253916 - cp's OEIS frontend

A253917 Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.

72, 738, 2758, 16777232, 1073741856, 282429536508, 95367431640650, 150094635296999148, 221073919720733357899812, 311973482284542371301330321821976098, 1329227995784915872903807060280344640, 85070591730234615865843651857942052992

Offset: 1

Alex Ratushnyak, Jan 18 2015

nonn

Intersection of A253913 and A253775.

			72 = 2^6+2+6 = 8^2+8,
738 = 3^6+3+6 = 9^3+9,
2758 = 52^2+52+2 = 14^3+14,
16777232 = 4^12+4+12 = 8^8+8,
1073741856 = 2^30+2+30 = 32^6+32,
282429536508 = 3^24+3+24 = 27^8+27,
95367431640650 = 5^20+5+20 = 25^10+25,
150094635296999148 = 9^18+9+18 = 27^12+27,
221073919720733357899812 = 6^30+6+30 = 30^15+36,
311973482284542371301330321821976098 = 7^42+7+42 = 49^21+49,
1329227995784915872903807060280344640 = 4^60+4+60 = 64^20+64,
85070591730234615865843651857942052992 = 2^126+2+126 = 128^18+128,
etc. - _Robert G. Wilson v_, Jan 19 2015

Robert G. Wilson v, Table of n, a(n) for n = 1..17
Robert G. Wilson v, 51 identified terms with no assurance that these are the only terms less than 10^1000.

Cf. A253913, A253775, A253914, A253916.

f[n_] := Block[{t = Transpose@ Flatten[ Table[{m^k + m, m^k + m + k}, {m, 2, Floor@ Sqrt[2^n]}, {k, Floor@ Log[m, 2^(n - 1)] + 1, Floor@ Log[m, 2^n]}], 1]}, Intersection[ t[[1]], t[[2]]]]; f[1] = {}; Array[f, 50] // Flatten (* Robert G. Wilson v, Jan 19 2015 *)

a(7)-a(12) from Robert G. Wilson v, Jan 19 2015

A254034 Numbers representable as both b^c + b + c and x*y + x + y, where b, c, x, y are integers bigger than 1.

Original entry on oeis.org

8, 14, 32, 39, 44, 71, 74, 92, 134, 137, 158, 184, 212, 242, 251, 264, 266, 274, 308, 344, 353, 422, 464, 523, 554, 602, 634, 704, 741, 758, 814, 872, 932, 994, 1013, 1033, 1036, 1058, 1124, 1262, 1334, 1484, 1562, 1642, 1724, 1743, 1808, 1894, 1982, 2072, 2164, 2197

Offset: 1

Alex Ratushnyak, Jan 23 2015

nonn

			a(1) = 8 = 2^2 + 2 + 2 = 2*2 + 2 + 2.
a(2) = 14 = 3^2 + 3 + 2 = 4*2 + 4 + 2.
a(3) = 32 = 5^2 + 5 + 2 = 10*2 + 10 + 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A253916, A253917.

N:= 10000: # to get all entries <= N
A1:= {seq(seq(b^c+b+c, c = 2 .. floor(log[b](N))), b = 2 .. floor(sqrt(N)))}:
filter2:= proc(x) local x1;
   x1:= x+1;
   x <= N and not isprime(x1) and not(x1::even and isprime(x1/2))
end proc:
sort(convert(select(filter2,A1),list)); # Robert Israel, Dec 28 2020

mx = 2200; t = Transpose@ Flatten[ Table[{x^y + x + y, x*y + x + y}, {x, 2, Floor@ Sqrt@ mx}, {y, 2, Floor[mx/x]}], 1]; Intersection[ t[[1]], t[[2]]] (* Robert G. Wilson v, Jan 23 2015 *)

cp's OEIS Frontend

A253917 Numbers that can be represented as both x^y + x and b^c + b + c, for some b, c, x, y > 1.

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